Tensor products of Minimal Holomorphic Representations

Abstract

LetD=G/KD=G/Kbe an irreducible bounded symmetric domain with genusppandHν(D)H^{\nu }(D)the weighted Bergman spaces of holomorphic functions forν>p−1\nu >p-1. The spacesHν(D)H^\nu (D)form unitary (projective) representations of the groupGGand have analytic continuation inν\nu; they give also unitary representations whenν\nuin the Wallach set, which consists of a continuous part and a discrete part ofrrpoints. The first non-trivial discrete pointν=a2\nu =\frac a2gives the minimal highest weight representation ofGG. We give the irreducible decomposition of tensor productHa2⊗Ha2¯H^{\frac a2}\otimes \overline {H^{\frac a2}}. As a consequence we discover some new spherical unitary representations ofGGand find the expansion of the corresponding spherical functions in terms of theKK-invariant (Jack symmetric) polynomials, the coefficients being continuous dual Hahn polynomials.